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Holt McDougal Geometry
7-2 Similarity and Transformations 7-2 Similarity and Transformations
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
Holt McDougal Geometry
7-2 Similarity and Transformations
Warm Up
Find the image point when the indicated transformation is applied to the given pre-image point.
1. (x, y) → (x + 3, y – 1); (2, 4) (5, 3)
2. (x, y) → (x, –y); (–2, 1) (-2,-1)
3. (x, y) → (3x, 3y); (–5, 0) (-15,-0)
(1,-2) 4. (x, y) → ; (3, -6) 3 1 x,
3 1 y
Holt McDougal Geometry
7-2 Similarity and Transformations
Draw and describe similarity transformations in the coordinate plane.
Use properties of similarity transformations to determine whether polygons are similar and to prove circles similar.
Objectives
Holt McDougal Geometry
7-2 Similarity and Transformations
similarity transformation
Vocabulary
Holt McDougal Geometry
7-2 Similarity and Transformations
A transformation that produces similar figures is a similarity transformation. A similarity transformation is a dilation or a composite of one or more dilations and one or more congruence transformations. Two figures are similar if and only if there is a similarity transformation that maps one figure to the other figure.
Holt McDougal Geometry
7-2 Similarity and Transformations
Translations, reflections, and rotations are congruence transformations.
Remember!
Holt McDougal Geometry
7-2 Similarity and Transformations
Example 1: Drawing and Describing Dilations
D: (x, y) → (3x, 3y)
A(1, 1), B(3, 1), C(3, 2)
dilation with center (0, 0) and scale factor 3
A. Apply the dilation D to the polygon with the given vertices. Describe the dilation.
Holt McDougal Geometry
7-2 Similarity and Transformations
Example 1: Continued
dilation with center (0, 0) and scale factor 3 4
D: (x, y) →
P(–8, 4), Q(–4, 8), R(4, 4)
4 3 x,
4 3 y
B. Apply the dilation D to the polygon with the given vertices. Describe the dilation.
Holt McDougal Geometry
7-2 Similarity and Transformations
Check It Out! Example 1
D'(-2, 0), E'(-2, -1), F'(-1, -2); dilation with center
(0, 0) and scale factor 1 4
Apply the dilation D : (x, y)→
polygon with vertices D(-8, 0), E(-8, -4), and F(-4, -8). Name the coordinates of the image points. Describe the dilation.
to the 4 1 x,
4 1 y
Holt McDougal Geometry
7-2 Similarity and Transformations
A. A(–6, -6), B(-6, 3), C(3, 3),
D(3, -6) and H(-2, -2),
J(-2, 1), K(1, 1), L(1, -2)
Example 2 : Determining Whether Polygons are
Similar
Yes; ABCD maps to HJKL by a dilation:
(x, y) → 1
3 x
1
3 y ,
Determine whether the polygons with the given vertices are similar.
Holt McDougal Geometry
7-2 Similarity and Transformations
B. P(2, 0), Q(2, 4), R(4, 4),
S(4, 0) and W(5, 0),
X(5, 10), Y(8, 10), Z(8, 0).
Example 2: Continued
No; (x, y) → (2.5x, 2.5y) maps P to W, but not S to Z.
Holt McDougal Geometry
7-2 Similarity and Transformations
Example 2: Continued
C. A(1, 2), B(2, 2), C(1, 4) and
D(4, -6), E(6, -6), F(4, -2)
Yes; ABC maps to A’B’C’ by a translation: (x, y) → (x + 1, y - 5). Then A’B’C’ maps to DEF by a dilation: (x, y) → (2x, 2y).
Holt McDougal Geometry
7-2 Similarity and Transformations
D. F(3, 3), G(3, 6), H(9, 3),
J(9, –3) and S(–1, 1),
T(–1, 2), U(–3, 1), V(–3, –1).
Example 2: Continued
Yes; FGHJ maps to F’G’H’J’ by a reflection : (x, y) → (-x, y). Then F’G’H’J’ maps to STUV by a
dilation:
(x, y) 1
3 x
1
3 y ,
Holt McDougal Geometry
7-2 Similarity and Transformations
Determine whether the polygons with the given vertices are similar : A(2, -1), B(3, -1), C(3, -4) and P(3, 6), Q(3, 9), R(12, 9).
Check It Out! Example 2
The triangles are similar because ABC can be mapped to A'B'C' by a rotation: (x, y) → (-y, x),
and then A'B'C' can be mapped to PQR by a dilation: (x, y) → (3x, 3y).
Holt McDougal Geometry
7-2 Similarity and Transformations
Example 3: Proving Circles Similar
A. Circle A with center (0, 0) and radius 1 is similar to circle B with center (0, 6) and radius 3.
Holt McDougal Geometry
7-2 Similarity and Transformations
Example 3: Continued
Circle A can be mapped to circle A’ by a translation: (x, y) → (x, y + 6). Circle A’ and circle
B both have center (0, 6). Then circle A’ can be mapped to circle B by a dilation with center (0, 6) and scale factor 3. So circle A and circle B are similar.
B. Circle C with center (0, –3) and radius 2 is similar to circle D with center (5, 1) and radius 5.
Holt McDougal Geometry
7-2 Similarity and Transformations
Example 3: Continued
Circle C can be mapped to circle C’ by a translation: (x, y) → (x + 5, y + 4). Circle C’ and
circle D both have center (5, 1).Then circle C’ can be mapped to circle D by a dilation with center (5, 1) and scale factor 2.5. So circle C and circle D are similar.
Holt McDougal Geometry
7-2 Similarity and Transformations
Prove that circle A with center (2, 1) and radius 4 is similar to circle B with center (-1, -1) and radius 2.
Check It Out! Example 3
Circle A can be mapped to circle A' by a
translation: (x, y) → (x - 3, y - 2). Then circle A'
can be mapped to circle B by a dilation with center
(-1, -1) and scale factor So, circles A and B are
similar.
1 2
Holt McDougal Geometry
7-2 Similarity and Transformations
Eric wants to make a drawing to show how he will arrange three rugs in his room.
The large rug is twice the size of each small rug. Eric will first draw the small rug in the lower left corner and then the large rug in the middle. How can he draw those rugs?
Example 4: Application
Holt McDougal Geometry
7-2 Similarity and Transformations
Place the lower left rug in the first quadrant of a coordinate plane in a convenient position. Apply the dilation with center (0, 0) and scale factor 2: (x, y) → (2x, 2y).
Example 4: Continue
Holt McDougal Geometry
7-2 Similarity and Transformations
What if…? How could Tia draw the middle flag to make it 4 times the size of each of the other flags?
Check It Out! Example 4
Apply the dilation with center (0, 0) and scale
factor 4: (x, y) → (4x, 4y).
Holt McDougal Geometry
7-2 Similarity and Transformations
Lesson Quiz : Part-I
1. Apply the dilation D: (x, y) to the
polygon with vertices A(2, 4), B(2, 6), and C(6, 4). Name the coordinates of the image points. Describe the dilation.
A’(3, 6), B’(3, 9), C’(9, 6); dilation with center
(0, 0) and scale factor 3
2
Holt McDougal Geometry
7-2 Similarity and Transformations
Lesson Quiz : Part-II
No; (x, y) → (0.5x, 0.5y) maps A to P, but not B to Q.
Determine whether the polygons with the given vertices are similar.
2. A(-4, 4), B(6, 4), C(6, -4),
D(-4, -4) and P(-2, 2),
Q(4, 2), R(4, -2), S(-2, -2)
3. A(2, 2), B(2, 4), C(6, 4) and
D(3, -3), E(3, -6), F(9, -6)
Yes; △ ABC maps to △ A’B’C’ by a reflection: (x, y) → (x, -y). Then △ A’B’C’ maps to △DEF by a dilation:(x, y) → (1.5x, 1.5y).
Holt McDougal Geometry
7-2 Similarity and Transformations
Lesson Quiz : Part-III
4. Prove that circle A with center (0, 4) and radius 4 is similar to circle B with center (-2, -7) and radius 6.
Circle A can be mapped to circle A’ by a translation:
(x, y) → (x - 2, y - 11). Circle A’ and circle B both
have center (-2, -7). Then circle A’ can be mapped to
circle B by a dilation with center (-2, -7) and scale
factor 3 2
So circle A and circle B are similar.